#227105

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

716 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(6,11)(7,9)(12,19)(13,17)(14,18)(15,20)(21,22,25,27)(23,30,32,36)(24,29,26,33)(28,34,35,31);(1,3,4,2)(5,6,10,13)(7,11,9,17)(8,15,16,19)(12,18,20,14)(22,27)(28,35)(29,33)(30,36);(5,7,10,9)(6,11,13,17)(8,14,16,18)(12,19,20,15)(21,23,25,32)(22,28,27,35)(24,31,26,34)(29,36,33,30);(5,8,10,16)(6,12,13,20)(7,14,9,18)(11,19,17,15)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,36,35,30)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134301);